We define \(\Delta ^1_1=\Sigma ^1_1\cap \Pi ^1_1\) .
The pointclasses \(\Sigma ^1_1\)  and \(\Pi ^1_1\)  are closed under countable unions and intersections
and so contain all the Borel sets.
A fundamental result of Suslin (see [1]}) says that in any
Polish space \(\mathcal {B}(X)=\Delta ^1_1\) . One can continue to define the
\(\Sigma ^1_n\)  and \(\Pi ^1_n\)  sets for \(n \ge 2\) , forming the projective hierarchy,
but we will not consider any
classes beyond \(\Pi ^1_1\) and \(\Sigma ^1_1\)  in this paper.
